Impact-induced viscoelastic bungee-jumper jets with uniform extension and stress

Abstract

We investigate the dynamics of a "bungee-jumper" jet induced by an impulsive force, which retracts after reaching its peak extension. Despite the strongly extensional and highly nonequilibrium nature of this motion, the jet exhibits simple and uniform rheological responses. To elucidate its extensional behavior in a highly extensional regime quantified by large Deborah and Reynolds numbers ($De \approx 2.1 \times 10^1 - 3.3 \times 10^3$, $Re \approx 2.8 \times 10^1 - 4.6 \times 10^2$), we use high-speed velocimetry and polarization-based stress imaging to measure the spatial distribution of velocity and stress throughout jets made of dilute polyethylene oxide (PEO) solutions. The bungee-jumper jets are found to exhibit two uniform characteristics despite the extreme $De$ conditions: a consistent spatial distribution of the extensional rate and a nearly uniform stress distribution during the jetting motion. These uniformities indicate that the seemingly complex jet dynamics can in fact be effectively represented using a constitutive model with spatially uniform coefficients. Comparison of several viscoelastic models shows that the Voigt model provides the best agreement with the measured dynamics, while the single-spring model captures the essential behavior when elasticity dominates.

Figures (4)

• Figure 1: Impact-induced liquid-jet generation using a glass syringe:kamamoto2021 (a) before impact; (b) just after impact. An impulsive acceleration applied to the container produces a highly focused liquid jet with an extremely large extensional rate, enabling access to flow regimes that are difficult to achieve with conventional jetting or rheometric techniques.
• Figure 2: Formation and dynamics of bungee-jumper jets for different PEO solutions. (a), (b), and (c) Jet shapes for 2M, 5M, and 8M PEO solutions at similar initial velocities ($U' \approx 5~\mathrm{m/s}$). In all cases, the jet reaches a maximum extension before retracting toward the nozzle, but the extent and timing of retraction differ markedly across molecular weights (see Movie S1 in the supplementary material). (d) Time evolution of jet-tip displacement. Circles denote experimental data; green dotted and green solid curves represent fits by the single-spring and Voigt models, respectively. Orange dot-dashed and purple dashed curves show approximate predictions by the Maxwell and FENE–CR models. Across the range of viscoelasticities examined, the Voigt model provides the most consistent description, capturing both weak- and strong-elasticity cases, whereas the single-spring model is valid only when elasticity dominates (5M and 8M).
• Figure 3: Velocity and stress fields of the bungee-jumper jet (PEO 5M) at its maximum extension. (a) Velocity field obtained from high-speed velocimetry and (b) corresponding axial velocity profiles, showing a nearly linear variation with $y$, indicative of a uniform extensional rate. (c) Phase retardation $\Delta$ measured using high-speed polarization imaging and (d) birefringence $\Delta/D_{\mathrm{jet}}$ after correction for the jet diameter. Because phase retardation is related to the path-integrated normal stress through the stress–optic law,miyazaki2021yokoyama2023nakamine2024worby2024 the nearly uniform birefringence profile indicates that the extensional stress is also nearly uniform. Together, these results demonstrate two key uniformities of bungee-jumper jets: (i) a uniform extensional rate and (ii) a nearly uniform extensional stress along the jet.
• Figure 4: Comparison of viscous and elastic force contributions for the 2M PEO solution in the Voigt and FENE–CR models. The viscoelastic ratio quantifies the relative magnitude of viscous to elastic terms in each model. The Voigt model exhibits a substantially larger viscous contribution than the FENE–CR model, explaining its better agreement with experiments, even in regimes where elasticity remains significant. This trend supports the interpretation that viscosity plays an essential role in the extensional dynamics of bungee-jumper jets under the high-$De$, moderate-$Re$ conditions of this study.